The Independent Police Complaints Commission said on Friday it would not launch an immediate investigation into claims from the former cabinet minister Andrew Mitchell that the Metropolitan police had leaked parts of its "plebgate" inquiry to the media.

Mitchell had complained to the police watchdog last week after press reports said a Scotland Yard inquiry into the clash between the Tory MP and police at the Downing Street gates last September had found no evidence officers had lied.

The row cost Mitchell his job as chief whip after a police log was leaked to the media in which he was alleged to have called the officers "plebs". The controversy arose after Mitchell tried to ride his bicycle through the gates in Downing Street but was asked to dismount by the two officers on duty and walk through.

The Met was conducting an inquiry, supervised by the IPCC, into the leaking of the log. It was that inquiry which was allegedly the subject of another leak to the Guardian and the Times last week.

But the IPCC said the priority should be the original investigation. It did not rule out an inquiry into Mitchell's latest complaint at a later stage.

Deborah Glass, deputy chair of the IPCC, wrote to Mitchell saying: "While I fully understand your concerns about these press reports, it appears to me that the public interest is best served by ensuring that the MPS are indeed carrying out a robust and thorough investigation into the initial incident and its aftermath."

After last week's alleged leaks, Mitchell wrote to the IPCC saying: "We are deeply dismayed that the Metropolitan police appear to have leaked part of the report prepared for the Crown Prosecution Service to certain members of the press and spun it to the advantage of the police officers concerned."

Mitchell hopes the inquiry will help exonerate him and possibly allow his return to the Conservative frontbench.

The former Tory home affairs spokesperson, David Davis, who is helping Mitchell in his battle with the Met, said: "The reason for the complaint was that there appeared to be a spinning operation going on, emanating from within the Metropolitan police, which was putting a portrayal on the investigation which we did not recognise, which was designed to shape public perceptions. We thought that was quite wrong ahead of the proper judicial process."

Davis said Mitchell had no plans to take up the Met's offer to air his concerns to deputy assistant commissioner Pat Gallan, one of Met commissioner's Sir Bernard Hogan-Howe's most trusted aides. Davis said: "We won't be taking up the suggestion we see Pat Gallan at this stage."

The IPCC letter said there were several possible sources for last week's alleged leak. It added that further material related to the original inquiry needed to be reviewed by the IPCC and the CPS and it was hoped that task would be completed in the next two weeks.

Glass told Mitchell, the MP for Sutton Coldfield: "The CPS will make the final decision in relation to whether there is sufficient evidence and in the public interest to charge anyone.

"I also understand that the continued passage of time before this situation is resolved, during which speculation will inevitably persist, is extremely frustrating for you. I can only assure you that I am doing all I can to bring this to a fair conclusion – and that the relevant evidence will be published as soon as it possibly can be." The Met referred inquiries to the IPCC but a spokesperson said there had been no inquiry into last week's alleged leaks.